When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged (special-functions) or a discussion around some function reminding of the Weierstrass no where differentiable continuous function. These examples make me think - what would classify a special function?
A vague bad definition could be "A function is a special function if it has some resemblance to some Hypergeometric function" or "A function is a special function if it fits into the Bateman manuscript project.
To me the Gamma function and the Zeta function are definitively special functions.
Also, I have worked on Legendre functions $P_\lambda$ and $Q_\lambda$ of the first and second kind , which I would call special functions, but not individually however (except perhaps $P_{-1/2}$).
I would not say that elementary functions (such as trigonometric functions and the exponential function) are special functions - but I am not totally convinced about this..
I do not agree with Wikipedia, it says: "Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.
There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions are also considered as special functions."
Also, looking at the Wikipedia list (linked above) the indicator function, step functions, the absolute value function and the sign are special functions -- this sounds very wrong to me.
So what is a special function and what should be under the (special-function) tag?